Question:

For $ab + cd$ to be even, both $ab$ and $cd$ must have the same parity (both even or both odd). A two-digit number is even iff its units digit is even, and odd iff its units digit is odd. So we need $b$ and $d$ to have the same parity. Case 1: Both even - choose 2 distinct even digits from {2,4,6,8} for positions $b,d$ in $2 imes 1$ ways (ordered), then choose 2 distinct odd digits from {1,3,5,7,9} for positions $a,c$ in $5 imes 4$ ways. This gives $4 imes 3 imes 5 imes 4 = 240$ numbers. Case 2: Both odd - choose 2 distinct odd digits for $b,d$ in $5 imes 4$ ways, then choose 2 distinct even digits from {2,4,6,8} for $a,c$ in $4 imes 3$ ways. This gives $5 imes 4 imes 4 imes 3 = 240$ numbers. Total = $240 + 240 = 480 = 3 imes 160 = 4 imes 120 = 11 imes 43 + 7$. So $480$ is divisible by $3, 4,$ and $11$.